Theorem symgextfv 17838

Description: The function value of the extension of a permutation, fixing the additional element, for elements in the original domain. (Contributed by AV, 6-Jan-2019.)

Hypotheses

Ref	Expression
symgext.s	|- S = ( Base ` ( SymGrp ` ( N \ { K } ) ) )
symgext.e	|- E = ( x e. N |-> if ( x = K , K , ( Z ` x ) ) )

Assertion

Ref	Expression
symgextfv	|- ( ( K e. N /\ Z e. S ) -> ( X e. ( N \ { K } ) -> ( E ` X ) = ( Z ` X ) ) )

Distinct variable groups:   x, K   x, N   x, S   x, Z   x, X

Allowed substitution hint:   E( x)

Proof of Theorem symgextfv

Step	Hyp	Ref	Expression
1		eldifi 3732	. . . 4 |- ( X e. ( N \ { K } ) -> X e. N )
2		fvexd 6203	. . . . 5 |- ( ( K e. N /\ Z e. S ) -> ( Z ` X ) e. _V )
3		ifexg 4157	. . . . 5 |- ( ( K e. N /\ ( Z ` X ) e. _V ) -> if ( X = K , K , ( Z ` X ) ) e. _V )
4	2, 3	syldan 487	. . . 4 |- ( ( K e. N /\ Z e. S ) -> if ( X = K , K , ( Z ` X ) ) e. _V )
5		eqeq1 2626	. . . . . 6 |- ( x = X -> ( x = K <-> X = K ) )
6		fveq2 6191	. . . . . 6 |- ( x = X -> ( Z ` x ) = ( Z ` X ) )
7	5, 6	ifbieq2d 4111	. . . . 5 |- ( x = X -> if ( x = K , K , ( Z ` x ) ) = if ( X = K , K , ( Z ` X ) ) )
8		symgext.e	. . . . 5 |- E = ( x e. N |-> if ( x = K , K , ( Z ` x ) ) )
9	7, 8	fvmptg 6280	. . . 4 |- ( ( X e. N /\ if ( X = K , K , ( Z ` X ) ) e. _V ) -> ( E ` X ) = if ( X = K , K , ( Z ` X ) ) )
10	1, 4, 9	syl2anr 495	. . 3 |- ( ( ( K e. N /\ Z e. S ) /\ X e. ( N \ { K } ) ) -> ( E ` X ) = if ( X = K , K , ( Z ` X ) ) )
11		eldifsn 4317	. . . . . 6 |- ( X e. ( N \ { K } ) <-> ( X e. N /\ X =/= K ) )
12		df-ne 2795	. . . . . . . 8 |- ( X =/= K <-> -. X = K )
13	12	biimpi 206	. . . . . . 7 |- ( X =/= K -> -. X = K )
14	13	adantl 482	. . . . . 6 |- ( ( X e. N /\ X =/= K ) -> -. X = K )
15	11, 14	sylbi 207	. . . . 5 |- ( X e. ( N \ { K } ) -> -. X = K )
16	15	adantl 482	. . . 4 |- ( ( ( K e. N /\ Z e. S ) /\ X e. ( N \ { K } ) ) -> -. X = K )
17	16	iffalsed 4097	. . 3 |- ( ( ( K e. N /\ Z e. S ) /\ X e. ( N \ { K } ) ) -> if ( X = K , K , ( Z ` X ) ) = ( Z ` X ) )
18	10, 17	eqtrd 2656	. 2 |- ( ( ( K e. N /\ Z e. S ) /\ X e. ( N \ { K } ) ) -> ( E ` X ) = ( Z ` X ) )
19	18	ex 450	1 |- ( ( K e. N /\ Z e. S ) -> ( X e. ( N \ { K } ) -> ( E ` X ) = ( Z ` X ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   ifcif 4086   {csn 4177   |-> cmpt 4729   ` cfv 5888   Basecbs 15857   SymGrpcsymg 17797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  symgextf1lem  17840  symgextf1  17841  symgextfo  17842  symgextres  17845